Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ a&gt;0,b&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ V(x)\geq0 $\end{document}</tex-math></inline-formula> be a coercive function in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^2 $\end{document}</tex-math></inline-formula>. We study the following constrained minimization problem on a suitable weighted Sobolev space <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula>: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} e_{a}(b): = \inf\left\{E_{a}^{b}(u):u\in\mathcal{H}\ \mbox{and}\ \int_{\mathbb R^{2}}|u|^{2}dx = 1\right\}, \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M5">\begin{document}$ E_{a}^{b}(u) $\end{document}</tex-math></inline-formula> is a Kirchhoff type energy functional defined on <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula> by <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} E_{a}^{b}(u) = \frac{1}{2}\int_{\mathbb R^{2}}[|\nabla u|^{2}+V(x)u^{2}]dx+\frac{b}{4}\left(\int_{\mathbb R^{2}}|\nabla u|^{2}dx\right)^{2}-\frac{a}{4}\int_{\mathbb R^{2}}|u|^{4}dx. \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>It is known that, for some <inline-formula><tex-math id="M7">\begin{document}$ a^{\ast}&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ e_{a}(b) $\end{document}</tex-math></inline-formula> has no minimizer if <inline-formula><tex-math id="M9">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ a\geq a^{\ast} $\end{document}</tex-math></inline-formula>, but <inline-formula><tex-math id="M11">\begin{document}$ e_{a}(b) $\end{document}</tex-math></inline-formula> has always a minimizer for any <inline-formula><tex-math id="M12">\begin{document}$ a\geq0 $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M13">\begin{document}$ b&gt;0 $\end{document}</tex-math></inline-formula>. The aim of this paper is to investigate the limit behaviors of the minimizers of <inline-formula><tex-math id="M14">\begin{document}$ e_{a}(b) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M15">\begin{document}$ b\rightarrow0^{+} $\end{document}</tex-math></inline-formula>. Moreover, the uniqueness of the minimizers of <inline-formula><tex-math id="M16">\begin{document}$ e_{a}(b) $\end{document}</tex-math></inline-formula> is also discussed for <inline-formula><tex-math id="M17">\begin{document}$ b $\end{document}</tex-math></inline-formula> close to 0.

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